Sharp threshold for the Ising perceptron model

Consider the discrete cube {−1,1}N and a random collection of half spaces which includes each space H(x):={y∈{−1,1}N:x·y≥κ N} for x∈{−1,1}N independently with probability p. Is intersection these empty? This is called Ising perceptron model under Bernoulli disorder. We prove that this event has sharp threshold, is, empty increases quickly from ϵ to 1−ϵ when p only by factor 1+o(1) as N→∞.